Find the value of `( x + a)/( x - a) + ( x + b)/( x - b) `, if ` x = (2ab)/( a + b)`

Options

A. `-2`
B. 2
C. 1
D. `-1`

Answer

To find the value of the expression \(\frac{x + a}{x - a} + \frac{x + b}{x - b}\) given that \(x = \frac{2ab}{a + b}\), we can follow these steps:

### Step 1: Substitute the value of \(x\)
Substituting \(x = \frac{2ab}{a + b}\) into the expression:

\[ \frac{\frac{2ab}{a + b} + a}{\frac{2ab}{a + b} - a} + \frac{\frac{2ab}{a + b} + b}{\frac{2ab}{a + b} - b} \]

### Step 2: Simplify \(\frac{x + a}{x - a}\)
Now, let's simplify the first part \(\frac{x + a}{x - a}\):

1. **Numerator**:
\[ x + a = \frac{2ab}{a + b} + a = \frac{2ab + a(a + b)}{a + b} = \frac{2ab + a^2 + ab}{a + b} = \frac{a^2 + 3ab}{a + b} \]

2. **Denominator**:
\[ x - a = \frac{2ab}{a + b} - a = \frac{2ab - a(a + b)}{a + b} = \frac{2ab - a^2 - ab}{a + b} = \frac{ab - a^2}{a + b} = \frac{a(b - a)}{a + b} \]

So,
\[ \frac{x + a}{x - a} = \frac{\frac{a^2 + 3ab}{a + b}}{\frac{a(b - a)}{a + b}} = \frac{a^2 + 3ab}{a(b - a)} \]

### Step 3: Simplify \(\frac{x + b}{x - b}\)
Now, let's simplify the second part \(\frac{x + b}{x - b}\):

1. **Numerator**:
\[ x + b = \frac{2ab}{a + b} + b = \frac{2ab + b(a + b)}{a + b} = \frac{2ab + ab + b^2}{a + b} = \frac{3ab + b^2}{a + b} \]

2. **Denominator**:
\[ x - b = \frac{2ab}{a + b} - b = \frac{2ab - b(a + b)}{a + b} = \frac{2ab - ab - b^2}{a + b} = \frac{ab - b^2}{a + b} = \frac{b(a - b)}{a + b} \]

So,
\[ \frac{x + b}{x - b} = \frac{\frac{3ab + b^2}{a + b}}{\frac{b(a - b)}{a + b}} = \frac{3ab + b^2}{b(a - b)} \]

### Step 4: Combine the two parts
Now, we can combine both parts:

\[ \frac{x + a}{x - a} + \frac{x + b}{x - b} = \frac{a^2 + 3ab}{a(b - a)} + \frac{3ab + b^2}{b(a - b)} \]

### Step 5: Find a common denominator
The common denominator is \(ab(b - a)\):

\[ = \frac{(a^2 + 3ab)b}{ab(b - a)} + \frac{(3ab + b^2)a}{ab(b - a)} \]

### Step 6: Combine the numerators
Combining the numerators:

\[ = \frac{(a^2b + 3ab^2) + (3a^2b + ab^2)}{ab(b - a)} = \frac{4a^2b + 4ab^2}{ab(b - a)} = \frac{4ab(a + b)}{ab(b - a)} = \frac{4(a + b)}{b - a} \]

### Final Result
Thus, the value of the expression is:

\[ \frac{4(a + b)}{b - a} \]